Estimating the Reproductive Number of COVID-19
Louis Rossouw
2020-07-25
1 Introduction
This paper contains estimates for the reproductive number \(R_{t,m}\) over time \(t\) in various countries \(m\). It also shows the same for provinces within South Africa.. This is done using the methodology as described in [1]. These have been implemented in R using EpiEstim package [2] which is what is used here.
This paper and it’s results should be updated roughly daily and is available online.
2 Updates
As this paper is updated over time this section will summarise significant changes. The code producing this paper is tracked using Git. The Git commit hash for this project at the time of generating this paper was 88bbcdc30bb19d4bcb71d8409a5c8330787cac00.
2020-06-12
- Initial version.
2020-06-13
- Combine the version for South Africa and other countries.
2020-06-14
- Add Google Mobility data for comparison and further modelling.
2020-06-15
- Add more details about methodology.
2020-06-17
- Tidy-up to remove the code.
3 Countries
All countries available in the data will be analysed but also the provinces of South Africa. This paper produces charts for the following countries only:
- Brazil
- Canada
- Chile
- India
- Ireland
- Italy
- Peru
- South Africa
- Spain
- United Kingdom
4 Data
4.1 South Africa
Data is downloaded from the Git repository associated with [3].
In the case data file row 21 and 32 contain no provincial details. We estimated it by spreading the national total to the provinces in proportion to the combined mixture of the prior day and the next day.
Further fixes are applied to both case and death data:
- Scale up the per province data for unknown values.
- This results in provincial data which are not whole numbers. These are rounded to the nearest whole number.
- A
SAcolumn is added as the sum of the new per province data. - Data is formatted and disaggregated such that item represents the incremental cases or deaths rather than cumulative figures.
- Data is filled with data (albeit with 0 cases or deaths) for all dates in the range.
- Any incremental case or death counts that are negative are zeroeised.
- New cumulative figures are calculated.
4.2 Other countries
Data for other countries are downloaded from [4]. Minor formatting is applied to get the data ready for further processing.
For Spain death data reporting of deaths appear to have significantly changed and we therefore exclude the data from this analysis [5].
4.3 Google Mobility Data
Google released mobility data indexes that deviations from a baseline of movement during 3 January to 6 February 2020 [6]. This data is downloaded and prepared for linking to our estimates of \(R_{t,m}\). The data is made available as a comma-separated file which is updated regularly but not daily. Data is sometimes up to a week behind. For countries other than South Africa only the country level mobility data is kept. I.e. we ignore regional data within countries. For South Africa provincial data is also kept.
We also calculate an average mobility index per [7]. This index averages the mobility indexes but exclude the following indexes:
- Residential index is excluded as it’s less likely to drive the epidemic (and is inversely correlated),
- Mobility in parks is also unlikely to be a driving factor.
5 Basic Exploration
5.1 South Africa
Below we plot cumulative case count on a log scale by province:
Below we plot the cumulative deaths by province on a log scale:
Below we plot average mobility indexes by province:
5.2 Other Countries
Below we plot cumulative case count on a log scale by country:
Below we plot the cumulative deaths by country on a log scale:
Below we plot average mobility indexes by country:
6 Method & Assumptions
This analysis follows the follows the method proposed [1].
Essentially this models the following relationship:
\(E(I_{t,m})=R_{t,m}\sum_{s=1}^tI_{t-s,m}w_{s}\) where:
- \(m\) is the country of province in case of South African data.
- \(I_{t,m}\) is the number of cases (or deaths) reported at time \(t\).
- \(R_{t,m}\) is the “instantaneous reproduction number” to quote [1].
- \(I_{t,m} \sim Poisson\) with the above mean.
- \(w_{s}\) is the infectivity profile of an individual over time \(s\).
The formulation of instantaneous rate of transmission proposed in [1] is convenient because it ensures that \(R_{t,m}\) only depends on information that is known at time \(t\). We do not need to know about future transmission.
6.1 Serial Interval
To do further analysis an serial interval assumption is needed (\(w_{s}\)). The serial interval is taken from [8]. It’s assumed to be Gamma distributed with mean of 6.48 and standard deviation of 3.83. This corresponds to the effective infectiousness of an individual since acquiring the infection themselves.
We plot this serial distribution below:
6.2 Time Window Size
[1] describes a choice of a time window to estimate \(R_{t,m}\). Based on this analysis we chose a time window of 7-days (\(\tau\) using notation of [1]). During this window \(R_{t,m}\) is assumed to be constant. However information prior that is taken into account.
Note that the time window size does not interact with the serial interval as information prior to the window is taken into account. It’s just the period during which \(R_{t,m}\) is assumed to be constant.
Based on the table in Appendix 2 of [1] we believe a window of 7-days to be reasonable as long as that window contains 12 or more cases or deaths.
7 Estimate \(R_{t,m}\)
7.1 Estimation Routine
The analysis works backwards and fits \(R_{t,m}\) values for whole weeks (ending on the last date in the data) using the EpiEstim package.
So \(t\) is based on the date of reporting (be that cases or deaths) and \(m\) is the country or province. Two values are estimated for each country:
- \(R_{t,m}^{cases}\) is the reproductive number implied by the cases reported at time \(t\) in country \(m\).
- \(R_{t,m}^{deaths}\) is the reproductive number implied by the deaths reported at time \(t\) in country \(m\).
Note that the time periods are left unadjusted, though in reality the \(R_{t,m}^{deaths}\) should be shifted back approximately 2 weeks relative to \(R_{t,m}^{cases}\).
8 Linking \(R_{t,m}\) estimates to mobility
Linking the mobility data should be done in a way that corresponds to the time the infection should have occurred not the time of death. For cases we assume 6 days from infection to case being diagnosed plus a further 7-day reporting delay. For death we assume 23 days from infection to death and a further 7-day reporting delay as well.
Mobility data is then averaged over the weeks corresponding to the \(R_{t,m}\) shifted as above.
9 Modelling \(R_{t,m}\) as a function of Mobility Indexes
TO DO
10 Results
10.1 Current \(R_{t,m}\) estimates
Below current (last weekly) \(R_{t,m}\) estimates are tabulated.
10.1.1 South Africa
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| EC | cases | 11,792 | 2020-07-25 | 0.9 | 0.9 | 1.0 |
| EC | deaths | 639 | 2020-07-25 | 2.2 | 2.4 | 2.5 |
| FS | cases | 6,508 | 2020-07-25 | 1.4 | 1.4 | 1.5 |
| FS | deaths | 16 | 2020-07-25 | 0.4 | 0.7 | 1.0 |
| GP | cases | 28,210 | 2020-07-25 | 0.9 | 0.9 | 0.9 |
| GP | deaths | 579 | 2020-07-25 | 1.5 | 1.6 | 1.8 |
| KZN | cases | 20,445 | 2020-07-25 | 1.2 | 1.2 | 1.3 |
| KZN | deaths | 201 | 2020-07-25 | 1.3 | 1.5 | 1.7 |
| LP | cases | 2,242 | 2020-07-25 | 1.3 | 1.3 | 1.4 |
| LP | deaths | 0 | 2020-07-25 | 0.0 | 0.4 | 1.4 |
| MP | cases | 3,773 | 2020-07-25 | 1.3 | 1.3 | 1.4 |
| MP | deaths | 8 | 2020-07-25 | 0.2 | 0.4 | 0.7 |
| NC | cases | 1,192 | 2020-07-25 | 1.3 | 1.4 | 1.4 |
| NC | deaths | 12 | 2020-07-25 | 1.2 | 2.3 | 3.7 |
| NW | cases | 3,541 | 2020-07-25 | 0.9 | 1.0 | 1.0 |
| NW | deaths | 0 | 2020-07-25 | 0.0 | 0.1 | 0.4 |
| SA | cases | 83,319 | 2020-07-25 | 1.0 | 1.0 | 1.0 |
| SA | deaths | 1,707 | 2020-07-25 | 1.5 | 1.5 | 1.6 |
| WC | cases | 5,774 | 2020-07-25 | 0.8 | 0.8 | 0.8 |
| WC | deaths | 252 | 2020-07-25 | 0.8 | 0.9 | 1.0 |
10.1.2 Other Countries
The table below contains \(R_{t,m}\) estimates for the last week available. These estimates are based either on cases or deaths and a 95% confidence interval is shown.
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Brazil | cases | 297,038 | 2020-07-25 | 1.2 | 1.2 | 1.2 |
| Brazil | deaths | 7,387 | 2020-07-25 | 1.0 | 1.0 | 1.0 |
| Canada | cases | 3,535 | 2020-07-25 | 1.3 | 1.3 | 1.3 |
| Canada | deaths | 42 | 2020-07-25 | 0.5 | 0.6 | 0.8 |
| Chile | cases | 14,765 | 2020-07-25 | 0.9 | 0.9 | 0.9 |
| Chile | deaths | 567 | 2020-07-25 | 1.1 | 1.2 | 1.3 |
| India | cases | 298,145 | 2020-07-25 | 1.3 | 1.3 | 1.3 |
| India | deaths | 5,085 | 2020-07-25 | 1.2 | 1.2 | 1.3 |
| Ireland | cases | 115 | 2020-07-25 | 0.9 | 1.0 | 1.2 |
| Ireland | deaths | 11 | 2020-07-25 | 1.1 | 2.0 | 3.2 |
| Italy | cases | 1,623 | 2020-07-25 | 1.1 | 1.2 | 1.2 |
| Italy | deaths | 69 | 2020-07-25 | 0.6 | 0.8 | 0.9 |
| Peru | cases | 30,424 | 2020-07-25 | 1.1 | 1.1 | 1.2 |
| Peru | deaths | 5,044 | 2020-07-25 | 3.3 | 3.3 | 3.4 |
| South_Africa | cases | 84,402 | 2020-07-25 | 1.0 | 1.0 | 1.0 |
| South_Africa | deaths | 1,539 | 2020-07-25 | 1.4 | 1.5 | 1.6 |
| Spain | cases | 12,166 | 2020-07-24 | 1.8 | 1.8 | 1.8 |
| Sweden | cases | 1,716 | 2020-07-25 | 0.7 | 0.7 | 0.7 |
| Sweden | deaths | 78 | 2020-07-25 | 0.8 | 0.9 | 1.1 |
| United_Kingdom | cases | 4,675 | 2020-07-25 | 0.9 | 1.0 | 1.0 |
| United_Kingdom | deaths | 444 | 2020-07-25 | 0.9 | 0.9 | 1.0 |
10.1.3 Top 10 Countries
Below we show various extremes of \(R_{t,m}\) where counts (deaths or cases) exceed 50 in the last week.
10.1.3.1 Lowest \(R_{t,m}\) based on Deaths
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Pakistan | deaths | 265 | 2020-07-25 | 0.6 | 0.7 | 0.8 |
| Egypt | deaths | 330 | 2020-07-25 | 0.7 | 0.7 | 0.8 |
| Italy | deaths | 69 | 2020-07-25 | 0.6 | 0.8 | 0.9 |
| Afghanistan | deaths | 99 | 2020-07-25 | 0.6 | 0.8 | 0.9 |
| Philippines | deaths | 219 | 2020-07-25 | 0.7 | 0.8 | 0.9 |
| Serbia | deaths | 66 | 2020-07-25 | 0.7 | 0.9 | 1.1 |
| Guatemala | deaths | 226 | 2020-07-25 | 0.8 | 0.9 | 1.0 |
| Ecuador | deaths | 218 | 2020-07-25 | 0.8 | 0.9 | 1.0 |
| Russia | deaths | 799 | 2020-07-25 | 0.9 | 0.9 | 1.0 |
| Turkey | deaths | 122 | 2020-07-25 | 0.8 | 0.9 | 1.1 |
10.1.3.2 Lowest \(R_{t,m}\) based on Cases
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Sri_Lanka | cases | 67 | 2020-07-25 | 0.2 | 0.3 | 0.3 |
| Cameroon | cases | 551 | 2020-07-25 | 0.4 | 0.4 | 0.5 |
| Yemen | cases | 78 | 2020-07-25 | 0.5 | 0.6 | 0.7 |
| Pakistan | cases | 9,970 | 2020-07-25 | 0.7 | 0.7 | 0.7 |
| Mozambique | cases | 188 | 2020-07-25 | 0.6 | 0.7 | 0.8 |
| Sweden | cases | 1,716 | 2020-07-25 | 0.7 | 0.7 | 0.7 |
| Afghanistan | cases | 747 | 2020-07-25 | 0.7 | 0.7 | 0.8 |
| Egypt | cases | 4,598 | 2020-07-25 | 0.7 | 0.7 | 0.8 |
| Maldives | cases | 207 | 2020-07-25 | 0.7 | 0.8 | 0.9 |
| United_Arab_Emirates | cases | 1,827 | 2020-07-25 | 0.7 | 0.8 | 0.8 |
10.1.3.3 Highest \(R_{t,m}\) based on Deaths
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Peru | deaths | 5,044 | 2020-07-25 | 3.3 | 3.3 | 3.4 |
| Kyrgyzstan | deaths | 349 | 2020-07-25 | 1.6 | 1.7 | 1.8 |
| South_Africa | deaths | 1,539 | 2020-07-25 | 1.4 | 1.5 | 1.6 |
| Honduras | deaths | 226 | 2020-07-25 | 1.3 | 1.4 | 1.6 |
| Kazakhstan | deaths | 210 | 2020-07-25 | 1.2 | 1.4 | 1.6 |
| Israel | deaths | 56 | 2020-07-25 | 1.1 | 1.4 | 1.7 |
| Indonesia | deaths | 708 | 2020-07-25 | 1.3 | 1.4 | 1.5 |
| Kenya | deaths | 52 | 2020-07-25 | 1.0 | 1.3 | 1.7 |
| Argentina | deaths | 589 | 2020-07-25 | 1.2 | 1.3 | 1.4 |
| Colombia | deaths | 1,687 | 2020-07-25 | 1.2 | 1.2 | 1.3 |
10.1.3.4 Highest \(R_{t,m}\) based on Cases
| Country | Estimated Type | Count (Last Week) | Week Ending | R - Lower CI | R - Mean | R - Uppper CI |
|---|---|---|---|---|---|---|
| Sao_Tome_and_Principe | cases | 119 | 2020-07-25 | 6.7 | 8.1 | 9.6 |
| Bahamas | cases | 187 | 2020-07-25 | 3.8 | 4.4 | 5.0 |
| Gambia | cases | 138 | 2020-07-25 | 3.6 | 4.3 | 5.1 |
| Georgia | cases | 99 | 2020-07-25 | 1.9 | 2.4 | 2.8 |
| Uruguay | cases | 129 | 2020-07-25 | 2.0 | 2.3 | 2.7 |
| Finland | cases | 79 | 2020-07-25 | 1.8 | 2.2 | 2.7 |
| Gabon | cases | 669 | 2020-07-25 | 2.0 | 2.1 | 2.3 |
| China | cases | 799 | 2020-07-25 | 1.7 | 1.9 | 2.0 |
| Spain | cases | 12,166 | 2020-07-24 | 1.8 | 1.8 | 1.8 |
| Malaysia | cases | 106 | 2020-07-25 | 1.4 | 1.7 | 2.0 |
10.2 Result Plots
Below we plot results for each country/province in a list. We filter out weeks where the upper end of confidence interval for \(R_{t,m}\) exceeds five.
10.2.1 South Africa
Below we plot estimates for all the provinces. Note that [1] recommends only to start using estimates after at least 12 cases have occurred. On that basis we do not have estimates for R based on deaths for a number of provinces.
10.2.2 Other Countries
Below we plot all the countries:
10.2.3 Countries by Continent
10.2.3.1 Africa
10.2.3.2 America
10.2.3.3 Asia
